Landau's function for one million billions Marc Deleglise, Jean-Louis Nicolas and Paul Zimmermann ? February 27, 2008 A Henri Cohen pour son soixantieme anniversaire. Abstract Let Sn denote the symmetric group with n letters, and g(n) the max- imal order of an element of Sn. If the standard factorization of M into primes is M = q?11 q?22 . . . q?kk , we define ?(M) to be q ?1 1 + q?22 + . . .+ q?kk ; one century ago, E. Landau proved that g(n) = max?(M)≤n M and that, when n goes to infinity, log g(n) ? p n log(n). There exists a basic algorithm to compute g(n) for 1 ≤ n ≤ N ; its running time is O “ N3/2/ √ logN ” and the needed memory is O(N); it allows computing g(n) up to, say, one million. We describe an algorithm to calculate g(n) for n up to 1015. The main idea is to use the so-called ?-superchampion numbers. Similar numbers, the superior highly composite numbers, were introduced by S. Ramanujan to study large values of the divisor function ? (n) = Pd |n 1.

In the sequel of our article, “ step ” will refer to one of the above six steps, and “ the algorithm ” will refer to the algorithm sketched in Section 1.3. On the web site of the second author, there is a Maple code of this algorithm where each instruction is explained according with the notation of this article. If we want to calculate g ( n ) for consecutive values n = n 1  n = n 1 +1      n = n 2 , most of the operations of the algorithm are similar and can be put in com-mon; however, due to some technical questions, it is more diﬃcult to treat this problem, and here, we shall restrict ourselves to the computation of g ( n ) for one value of n . To compute the ﬁrst 5000 highly composite numbers, G. Robin (cf. [27]) already used a notion of beneﬁt similar to that introduced in this article. 1.4 The function G ( p k  m ) In step 6, the computation of the suﬃx of g ( n ) leads to the function G ( p k m ), deﬁned by Deﬁnition 1. Let p k be the k -th prime, for some k ≥ 3 and m an integer satisfying 0 ≤ m ≤ p k +1 − 3 . We deﬁne G ( p k  m ) = max Qq 11 Qq 22 qQ ss (1.12) where the maximum is taken over the primes Q 1  Q 2      Q s  q 1  q 2      q s ( s ≥ 0 ) satisfying 3 ≤ q s < q s − 1 <    < q 1 ≤ p k < p k +1 ≤ Q 1 < Q 2 <    < Q s (1.13) and s X ( Q i − q i ) ≤ m (1.14) i =1 This function G ( p k  m ) is interesting in itself. It satisﬁes ℓ ( G ( p k  m )) ≤ m (1.15) We study it in Section 8, where a combinatorial algorithm is given to compute its value when m is not too large. For m large, a better algorithm is given in Section 9. Let us denote by  1 ( n ) <  2 ( n ) <    the increasing sequence of the primes which do not divide g ( n ), and by P ( n ) the largest prime factor of g ( n ). It is shown in [17] that lim n →∞ P ( n )  1 ( n ) = 1. We may guess from Proposition 10 that  1 ( n ) can be much smaller than P ( n ) while  2 ( n ) is closer to P ( n ). It seems diﬃcult to prove any result in this direction. 1.5 The running time Though we have the feeling that the algorithm presented in this paper (and implemented in Maple ) yields the value of g ( n ) for all n ’s up to 10 15 (and eventually for greater n ’s) in a reasonable time, it is not proved to do so. Indeed, we do not know how to get an eﬀective upper bound for the beneﬁt of g ( n ) (see sections 6, 7.3 and 11.1) and in the second and third steps, what 4